On Poisson geometries related to noncommutative emergent gravity

We study metric-compatible Poisson structures in the semi-classical limit of noncommutative emergent gravity. Space–time is realized as quantized symplectic submanifold embedded in R^D${\mathbb{R}}^D$, whose effective metric depends on the embedding as well as on the Poisson structure. We study solutions of the equations of motion for the Poisson structure, focusing on a natural class of solutions such that the effective metric coincides with the embedding metric. This leads to i$i$-(anti-) self-dual complexified Poisson structures in four space–time dimensions with Lorentzian signature. Solutions on manifolds with conformally flat metric are obtained and tools are developed which allow to systematically re-derive previous results, e.g. for the Schwarzschild metric. It turns out that the effective gauge coupling is related to the symplectic volume density, and may vary significantly over space–time. To avoid this problem, we consider in a second part space–time manifolds with compactified extra dimensions and split noncommutativity, where solutions with constant gauge coupling are obtained for several physically relevant geometries.